6. For any two n x n non-singular matrices A and B, then(a) K(A. B) ≥ K(A). K(B)(c) K(A. B) < K(A). K(B)aO bOO dC(b) K(A. B) > K(A). K(B)(d) K(A. B) ≤ K(A). K(B)

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Answered: 6. For any two nx n non-singular…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.3: Subgroups

Problem 17E: 17. Consider the set of matrices, where , , , ...

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Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[ 1001 ] D=[ 0110 ] T=[ 0110 ] in GL(2,), and let G={ I2,R,R2,R3,H,D,V,T }. Given that G is a group of order 8 with respect to multiplication, write out a multiplication table for G. Sec. 3.3,22b,32b Find the center Z(G) for each of the following groups G. b. G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1. Find the centralizer for each element a in each of the following groups. b. G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1 Sec. 4.1,22 22. Find an isomorphism from the octic group D4 in Example 12 of this section to the group G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of Section 3.1. Sec. 4.6,14 14. Let G={ I2,R,R2,R3,H,D,V,T } be the multiplicative group of matrices in Exercise 36 of section 3.1, let G={ 1,1 } under multiplication, and define :GG by ([ abcd ])=adbc. Assume that is an epimorphism, and find the elements of K= ker . Write out the distinct elements of G/K. Let :G/KG be the isomorphism described in the proof of Theorem 4.27, and write out the values of .
17. Consider the set of matrices, where , , , , Show that is s subgroup of , the general linear group of order over .
1. Suppose that, in Example 2.27, 400 units of food A, 600 units of B, and 600 units of C are placed in the test tube each day and the data on daily food consumption by the bacteria (in units per day) are as shown in Table 2.6. How many bacteria of each strain can coexist in the test tube and consume all of the food? Table 2.6 Bacteria Strain I Bacteria Strain II Bacteria Strain III Food A 1 2 0 Food B 2 1 1 Food C 1 1 2

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6. For any two nx n non-singular matrices A and B, then (a) K(A. B) ≥ K(A). K(B) (c) K(A. B) < K(A). K(B) O a O b O O d (b) K(A. B) > K(A). K(B) (d) K(A. B) ≤ K(A). K(B)